Is the Isotropic Constant Scaling Invariant?

Question

The entries of the covariance matrix of a convex body $K$ are defined as \begin{equation} \label{last} (a_{ij}) = \frac{\int_K x_ix_j}{|K|} - \frac{\int_K x_i}{|K|}\frac{\int_K x_j}{|K|}. \end{equation} We define the isotropic constant of any convex body $K$ using \begin{equation} \label{last2} L^{2n}_{K} :=\frac{\text{Det}\text({Cov{K}})}{|K|^2}. \end{equation} Is $L_K$ scaling invariant? To me that determinant looks scaling invariant. If the isotropic constant is scaling invariant where I can find a proof?